Databases of quantum periods for Fano manifolds

Fano manifolds are basic building blocks in geometry – they are, in a precise sense, atomic pieces of shapes. The classification of Fano manifolds is therefore an important problem in geometry, which has been open since the 1930s. One can think of this as building a Periodic Table for shapes. A recent breakthrough in Fano classification involves a technique from theoretical physics called Mirror Symmetry. From this perspective, a Fano manifold is encoded by a sequence of integers: the coefficients of a power series called the regularized quantum period. Progress to date has been hindered by the fact that quantum periods require specialist expertise to compute, and descriptions of known Fano manifolds and their regularized quantum periods are incomplete and scattered in the literature. We describe databases of regularized quantum periods for Fano manifolds in dimensions up to four. The databases in dimensions one, two, and three are complete; the database in dimension four will be updated as new four-dimensional Fano manifolds are discovered and new regularized quantum periods computed.

G t c t ( ) 1 (1) www.nature.com/scientificdata www.nature.com/scientificdata/ constraint on their complex structure. In general r d is a rational number, because curves with a symmetry group of order n are counted with weight 1/n, but in all known cases the coefficients of G X are integers. Gromov-Witten invariants remain constant under deformation of X, so G X is a deformation invariant of X.
This paper describes databases of regularized quantum periods of Fano manifolds in dimensions one, two, three, and four [31][32][33][34] . The databases were prepared by aggregating and standardising existing descriptions of regularized quantum periods in the literature, and computing the regularized quantum periods of various four-dimensional Fano manifolds where geometric constructions were known. The databases in dimensions up to three are complete; the database in dimension four will be updated as further four-dimensional Fano manifolds are discovered and new regularized quantum periods computed. This data will be useful to mathematicians and physicists who wish to identify Fano manifolds, to construct Calabi-Yau manifolds, or to investigate further the connection between Fano manifolds and Mirror Symmetry.

Methods
Let X be a Fano manifold. General theory implies that there is a differential operator L lt D (2)  ; see e.g. [26,Theorem 4.3]. We normalise the operator L in (2) by taking the l k to be integers with no common factor and insisting that l M >0, where M is the index k such that the pair (m k , n k ) is lexicographically maximal. The identity ≡ LG 0 X translates into a recurrence relation for the coefficients c d in (1) which determines all of the c d from the first few of them, and typically just from c 0 and c 1 ; here and henceforth we write c 0 = 1 and c 1 = 0.
Each entry in our databases describes the regularized quantum period of a Fano manifold. Two regularized quantum periods are assumed to be distinct unless it has been proven that they agree (i.e. that all the infinitely many coefficients c d in (1) agree). For example, two period sequences are the same if they have been shown to have the same differential operator, or if the Fano manifolds they are derived from have been shown to be deformation equivalent. More precisely, an entry in one of the databases corresponds to a sequence c 0 , c 1 ,… of coefficients of the regularized quantum period (1), where in some cases only finitely many terms c d may be known, along with information about the geometric origins of this period sequence. In dimensions one, two, and three the classification of Fano manifolds is known, and each deformation class of Fano manifolds has a distinct regularized quantum period. Thus in dimensions one, two, and three we can also interpret each entry in our databases as corresponding to a deformation class of Fano manifolds. In dimension four the classification of Fano manifolds is unknown, and it is possible that there exist Fano manifolds X 1 and X 2 that have the same regularized quantum period but that are not deformation equivalent. (No such examples are known, in any dimension, and we expect that no such examples exist.) One should therefore take care to think of each entry in the four-dimensional database as representing a regularized quantum period sequence of a Fano manifold rather than a deformation class of Fano manifolds. It may happen that two period sequences in the four-dimensional database are later proven to be equal. When this happens, we will update the database as described in the 'Data records' section.
The following methods were used to compute regularized quantum periods and the differential operators that annihilate them.
Mirror symmetry and the Lairez algorithm. For X a smooth Fano toric variety or toric complete intersection, mirror constructions by Givental 35 and Hori-Vafa 36 give a Laurent polynomial f that corresponds to X under Mirror Symmetry -see e.g. 28, for a summary of this. Given such a Laurent polynomial, one can compute a differential operator L such that LG 0 X ≡ using Lairez's generalised Griffiths-Dwork algorithm 37 . These steps are implemented in the Fanosearch software library; see the 'Code availability' section.
Products. If the regularized quantum periods of Fano manifolds X 1 and X 2 are known then the regularized quantum period of the product X 1 × X 2 is determined by [ . One can then use the Lairez algorithm to compute a differential operator L such that ≡ × LG 0 X X 1 2 .
Numerical linear algebra. Given sufficiently many coefficients c 0 , …, c M in (1), one can find a recurrence relation satisfied by this sequence using linear algebra. This determines a candidate for the differential operator L in (2). We applied this method to the Strangeway fourfolds, where closed formulas for the c i are known 39 . When the number M of coefficients involved is large, the linear system that determines the recurrence relation becomes highly overdetermined and so we can be confident that the operator L is correct. a note on rigour. When computing differential operators (2), neither the approach based on the Lairez algorithm nor the approach based on numerical linear algebra provides a proof that LG 0 X ≡ . In the latter case this is because the method cannot do so; in the former case this is because of an implementation detail in Lairez's algorithm: certain calculations over  t ( ) are made using reductions to for randomly-chosen www.nature.com/scientificdata www.nature.com/scientificdata/ primes p followed by a reconstruction step, and there is a (very small) probability of erroneous reconstruction. In each case, however, the probability of error is tiny.
In more detail: the Lairez algorithm computes the differential operator L by first computing a certain connection matrix M with entries in  t ( ). This matrix determines L uniquely. In the implementation of the algorithm that we used, the matrix M is reconstructed from its reductions M i to  t ( ) p i for a sequence p 1 , …, p k of randomly-chosen 32-bit primes p i . The entries of M are recovered from the entries of M i using rational reconstruction 40 , where we increase the number k of primes until the reconstruction stabilises. Let us consider a heuristic estimate of the probability of a single coefficient  ∈ q in a single entry of the matrix M being erroneously reconstructed from its mod-p i reductions. Set m = p 1 p 2 ⋯p k−1 . We have m od for some , k and since the reconstruction stabilises it follows that B ≡ A + rm mod mp k for some integer r. There are p k possible choices for B given A and (since p k was chosen uniformly at random among 32-bit primes) it seems reasonable that, if the reconstruction at step k − 1 was erroneous, then all possibilities for B are equally likely. Thus the probability of erroneous stabilisation is 1/p k . This is just one coefficient among many. Assuming that erroneous reconstructions of individual coefficients are independent, and using worst-case sizes for M and degrees of entries in M gives a probability of erroneous reconstruction on the order of 10 −6 . Furthermore the operators that we found satisfy a number of stringent checks described in the 'Technical validation' section. It is reasonable to conclude that they are correct.
Dimension one. The Fano manifold 1  is toric, and corresponds under Mirror Symmetry to the Laurent polynomial x + x −1 . The database of regularized quantum periods for one-dimensional Fano manifolds 31 , which contains one record, was constructed from this Laurent polynomial using Lairez's algorithm. To cross-check, one can use Givental's mirror theorem 41 to compute the regularized quantum period, finding Dimension two. The database in dimension two 32 was constructed by applying the Givental/Hori-Vafa mirror construction and the Lairez algorithm to the models of two-dimensional Fano manifolds as toric complete intersections given in [38, §G].  28 . The electronic supplementary material for that paper also provides the regularized quantum periods and differential operators (2) in machine-readable form. These differential operators were normalised as discussed after Eq. (2) and added to the database.

Key Type and Value
id a positive integer specifying the ID of the entry period a sequence of non-negative integers giving the first few terms c d in (1) names a sequence of strings specifying names for the Fano manifold: see Tables 2 and 3 pf_coefficients a sequence of integers l k as in (2) pf_exponents a sequence of length-two sequences [m k , n k ] of positive integers as in (2) pf_proven a boolean, true if and only if the operator (2) is proven to annihilate G X notes a string containing further notes on the Fano manifold duplicate a positive integer specifying the ID of another entry in the same database www.nature.com/scientificdata www.nature.com/scientificdata/ Four-dimensional Fano manifolds with classical constructions. Ref. 39 computes regularized quantum period sequences and differential operators (2) for many four-dimensional Fano manifolds with classical constructions, including all four-dimensional Fano manifolds with Fano index greater than one. These differential operators were normalised according to our conventions and added to the database. That paper also gives regularized quantum period sequences, but not differential operators, for a number of other four-dimensional Fano manifolds: • four-dimensional Fano toric varieties; • products of lower-dimensional Fano manifolds; • the Strangeway fourfolds.
Toric varieties are toric complete intersections, so four-dimensional Fano toric varieties were already included in the database. Differential operators for products of lower dimensional Fano manifolds were computed using the methods discussed in the section 'Products' above. Differential operators for the Strangeway fourfolds were computed as discussed in the section 'Numerical linear algebra' above. These differential operators were then normalised as discussed after Eq. (2) and added to the database.
Four-dimensional quiver flag zero loci. Four-dimensional Fano manifolds that are quiver flag zero loci in Fano quiver flag varieties of dimension up to 8 were classified by Kalashnikov 29 , who also computed the coefficients c d in (1) for these for d ≤ 15. Kalashnikov partitions the four-dimensional Fano quiver flag zero loci into equivalence classes depending on the values of the coefficients (c 0 , …, c 15 ) and finds 749 equivalence classes. It is not known, however, whether quiver flag zero loci with the same (c 0 , …, c 15 ) are deformation equivalent, so we regard the regularized quantum period sequence represented by (c 0 , …, c 15 ) as coming from the four-dimensional Fano manifold specified as the representative of the equivalence class in [29, Appendix B, Table 1]. These representatives were added to the database.

Data Records
We provide four key-value databases, containing regularized quantum periods for Fano manifolds in dimensions one, two, three, and four. These databases have been committed to the public domain using a CC0 license. They are available from the open access data repository Zenodo [31][32][33][34] . Zenodo provides versioned DOIs, and new versions of the databases will be produced if the data needs to be updated -for example because new four-dimensional quantum periods are computed. Each database is presented as an ASCII text file, called 'smooth_fano_N.txt' where N is the appropriate dimension. Each line of a record in that file contains a key, followed by ':' , followed by a value. Records are separated by a single blank line. Table 1 describes the keys provided by the records in each database and their corresponding values, where we think of each database as a single key-value table. Keys are strings of lower-case characters. Each entry in each database has an ID, which is a positive integer. IDs are sequential and start from 1, but carry no meaning or information other than to identify that particular entry in that particular database. Each entry in each database also specifies a non-empty sequence of names. These names identify Fano manifolds with that regularized quantum period in various published (partial) classifications, and hence determine constructions of these Fano manifolds. Names of the form 'S1 x S2' , where S1 and S2 are names of Fano manifolds X 1 and X 2 , indicate that the corresponding Fano manifold is a product X 1 × X 2 . Fano manifolds can have many different names. The databases of regularized quantum period sequences for one-, two-, and three-dimensional Fano manifolds use names as in Table 2; the database of regularized quantum period sequences for four-dimensional Fano manifolds uses names as in Table 3, and as specified in the file 'README.txt' in the Zenodo dataset 34 .
The differential operator L in (2) is recorded as parallel sequences of coefficients l k and exponents (m k , n k ); the corresponding keys in the database are named 'pf_coefficients' and 'pf_exponents' to reflect the fact that under Mirror Symmetry L corresponds to a Picard-Fuchs operator. The databases in dimensions one, two, and three contain values for all keys except 'duplicate'; in particular this determines the differential operator L in (2), and  Table 2. Names of entries in the databases for dimensions one, two, and three.
www.nature.com/scientificdata www.nature.com/scientificdata/ thus all coefficients c d in the expansion (1) of G X . For a number of the entries in the four-dimensional database, the differential operator (2) is unknown; in this case the key-value pairs related to the differential operator are omitted from the database entry.
The databases contain a key 'pf_proven' , with boolean values, that has the value false for each entry in each database such that (2) is known. This reflects the fact that, as discussed above, at the time of writing it is not proven that the differential operators (2) recorded in the database actually annihilate the corresponding regularized quantum periods. If this situation changes in the future -for example, if the certificated version of the Lairez algorithm described in [37, §7.3] is implemented -then we will make new versions of the databases available.
The presence of the key 'duplicate' in an entry indicates that this entry E is a duplicate of another entry in the same database, with the indicated ID, and that the entry E will not change further as the database is updated. The key 'duplicate' is not (and will never be) present in any of the databases in dimensions one, two, or three; at the time of writing it is also not present in the database in dimension four, but it will be added in future updates to indicate when two entries in that database have been proven to coincide.

technical Validation
The records in our databases satisfy a number of consistency checks. These can be verified, for example, using the computational algebra system Magma 42 and the Fanosearch software library; see Table 4 for the relevant function names, and the software documentation for these functions for the arguments and parameters required. Firstly, the period sequences in each database are annihilated by the corresponding differential operators (2) whenever they are known. Secondly. the differential operators L from (2) are expected to be of Fuchsian type, that is, to have only regular singular points. This is an extremely delicate condition on the coefficients l k , and can be checked by exact computation; in particular calls to RamificationData(L) or RamificationDefect(L) in the Fanosearch Magma library will raise an error if L is not Fuchsian. All the differential operators in the databases in dimensions one, two, and three are Fuchsian. Checking this for some of the entries in the four-dimensional database is prohibitively expensive, because it involves linear algebra over the splitting field of the symbol of the differential operator and this symbol can be of very high degree, but all of the four-dimensional operators for which the calculation was possible are Fuchsian, and all of them have regular singularities at those singular points defined over number fields of low degree.
As a further check, the differential operator (2) is expected to be of low ramification in the following sense 26 . Let S 1 ⊂  be a finite set and → V P S \ 1 a local system. Fix a basepoint ∈  x S \ 1 . For s ∈ S, choose a small loop that winds once anticlockwise around s and connect it to x via a path, thereby making a loop γ s about s based at x. Let T : The ramification defect of  is the quantity , and hence have non-negative ramification defect. A local system of ramification defect zero is called extremal. The ramification (and respectively ramification defect) of a differential operator L is the ramification (and respectively ramification defect) of the local system of solutions Lf ≡ 0. All the differential operators in the databases in dimensions one, two, and three are of low ramification; indeed with the exception of the two-dimensional Fano manifolds with names 'dP(7)' and 'dP(8)' , which have ramification defect 1, all of these operators are extremal. Computing the ramification for some of the operators in the four-dimensional database is prohibitively expensive, for the same reason as before, but all of the four-dimensional operators for which the calculation was possible are of low ramification, and many of them extremal or of ramification defect 1.  Table 3. Names of entries in the database for dimension four.

author contributions
T.C. and A.K. jointly conceived and performed the research and data analysis, and wrote the manuscript.